Group invariance of global generalized solutions of nonlinear PDEs in Colombeau algebras and in the Dedekind order completion method

Abstract

In this thesis a theoretical framework is provided within which large classes of nonlinear Lie groups of transformations are defined on spaces of generalized functions which yield global generalized solutions for large classes of nonlinear partial differential equations (PDEs). Although these function spaces contain the L Schwartz distributions, this theory stays within finite dimensional manifolds. It is shown how Lie symmetry groups for classical solutions of nonlinear PDEs can be extended to symmetry groups for global generalized solutions. For the first time in the literature, applications are given of the nonlinear group invariance of global weak solutions of nonlinear PDEs, under transformations defined on the whole domains of definition of those solutions. The major difficulty with the study of Lie group invariance of -mlutions of nonlinear PDEs has been the absence of a theory for the existence of solutions for general nonlinear PDEs. Also, many of the known classical solutions of specific nonlinear PDEs are not C -smooth on the whole domain of definition of these PDEs, typically exhibiting singularities. In the last few decades, functional analytic methods have produced existence results concerning global generalized solutions for several particular classes of nonlinear PDEs. However, these generalized solutions are usually linear functionals, such as the L Schwartz distributions, making the study of nonlinear group invariance particularly difficult, even though such transformations arise naturally with nonlinear PDEs. Furthermore, these linear functionals are defined on infinite dimensional vector spaces, making the computation of their Lie symmetry groups highly nontrivial. All of the above-mentioned difficulties are bypassed by two recent nonlinear theories of generalized functions. The first, based on algebraic solution methods, was developed by E E Rosinger, and in a particular and central case, in the independent work of J F Colombeau. The second is the more powerful, as yet unpublished, Dedekind order completion method of M O berguggenberger and E E Rosinger. Particularly the algebraic method of Rosinger and the order completion method of Oberguggenberger and Rosinger, have made available global existence results for solutions of large classes of nonlinear PDEs. In addition, the group transformations of the relevant generalized function spaces can be defined in such a way as to stay within finite dimensional manifolds which are, in fact, the original spaces of independent and dependent variables of the PDEs. It is shown here how to extend the concept of projectable group actions from classical function spaces to generalized function spaces, specifically to Colombeau's algebra of generalized functions, and to the order completion context of Oberguggenberger and Rosinger. In the case of Colombeau algebras, the same is done for more general groups. The concepts of invariant solutions and symmetry groups are also extended to include global generalized solutions. Finally, in the case of the order completion method, examples are given of the nonlinear group invariance of delta wave solutions of semilinear hyperbolic equations with rough initial data, and of Riemann solvers of the nonlinear shock wave equation.